Embedded newform 76.5.l.a.43.15

• Label: 76.5.l.a.43.15
• Level: $76$
• Weight: $5$
• Character: 76.43
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(38\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			43.15
Character	\(\chi\)	\(=\)	76.43
Dual form			76.5.l.a.23.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47024 - 3.72000i) q^{2} +(-11.9688 - 2.11041i) q^{3} +(-11.6768 + 10.9386i) q^{4} +(19.6524 + 16.4903i) q^{5} +(9.74622 + 47.6266i) q^{6} +(0.422999 - 0.244219i) q^{7} +(57.8592 + 27.3552i) q^{8} +(62.6822 + 22.8144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 48 q^{4} - 12 q^{5} - 60 q^{6} - 3 q^{8} - 180 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47024	−	3.72000i	−0.367560	−	0.930000i
							\(3\)	−11.9688	−	2.11041i	−1.32986	−	0.234491i	−0.536840	−	0.843684i	\(-0.680382\pi\)
−0.793022	+	0.609193i	\(0.791493\pi\)
\(5\)	19.6524	+	16.4903i	0.786095	+	0.659612i	0.944776	−	0.327718i	\(-0.106280\pi\)
							−0.158681	+	0.987330i	\(0.550724\pi\)
\(7\)	0.422999	−	0.244219i	0.00863264	−	0.00498406i	−0.495677	−	0.868507i	\(-0.665080\pi\)
							0.504310	+	0.863523i	\(0.331747\pi\)
\(11\)	−44.7121	−	25.8145i	−0.369521	−	0.213343i	0.303728	−	0.952759i	\(-0.401769\pi\)
							−0.673249	+	0.739415i	\(0.735102\pi\)
\(13\)	−8.23706	−	46.7147i	−0.0487400	−	0.276418i	0.950691	−	0.310139i	\(-0.100375\pi\)
							−0.999431	+	0.0337205i	\(0.989264\pi\)
\(17\)	197.442	−	71.8629i	0.683190	−	0.248661i	0.0229732	−	0.999736i	\(-0.492687\pi\)
							0.660216	+	0.751075i	\(0.270465\pi\)
\(19\)	338.433	−	125.634i	0.937488	−	0.348017i
							\(23\)	117.519	+	140.054i	0.222153	+	0.264752i	0.865597	−	0.500741i	\(-0.166939\pi\)
−0.643443	+	0.765494i	\(0.722495\pi\)
\(29\)	699.726	+	254.680i	0.832017	+	0.302829i	0.722686	−	0.691176i	\(-0.242907\pi\)
							0.109331	+	0.994005i	\(0.465129\pi\)
\(31\)	917.213	−	529.553i	0.954436	−	0.551044i	0.0599799	−	0.998200i	\(-0.480896\pi\)
							0.894456	+	0.447156i	\(0.147563\pi\)
\(37\)	2476.68			1.80911			0.904557	−	0.426352i	\(-0.140202\pi\)
							0.904557	+	0.426352i	\(0.140202\pi\)
\(41\)	447.585	−	2538.38i	0.266261	−	1.51004i	−0.499158	−	0.866511i	\(-0.666357\pi\)
							0.765419	−	0.643532i	\(-0.222532\pi\)
\(43\)	−1201.03	+	1431.33i	−0.649557	+	0.774112i	−0.985847	−	0.167646i	\(-0.946383\pi\)
							0.336290	+	0.941759i	\(0.390828\pi\)
\(47\)	−865.351	+	2377.53i	−0.391739	+	1.07629i	0.574468	+	0.818527i	\(0.305209\pi\)
							−0.966207	+	0.257767i	\(0.917013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.l.a.43.15	yes	228
4.3	odd	2	inner	76.5.l.a.43.23	yes	228
19.4	even	9	inner	76.5.l.a.23.23	yes	228
76.23	odd	18	inner	76.5.l.a.23.15	✓	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.l.a.23.15	✓	228	76.23	odd	18	inner
76.5.l.a.23.23	yes	228	19.4	even	9	inner
76.5.l.a.43.15	yes	228	1.1	even	1	trivial
76.5.l.a.43.23	yes	228	4.3	odd	2	inner